Matrix Methods 1st Midterm Definitions

Singular Matrix

A square matrix that does not have an inverse, often because its determinant is zero

Non-Singular Matrix

A matrix that has an inverse and is not singular, meaning its determinant is non-zero

Skew Symmetric

matrix A such that A^T = -A, meaning its transpose is equal to its negative

Linearly Independent

A set of vectors in a vector space that cannot be expressed as a linear combination of each other, meaning no vector in the set can be written as a combination of the others

Linearly Dependent

A set of vectors in a vector space that can be expressed as a linear combination of each other, meaning at least one vector in the set can be written as a combination of the others

Subspace

A matrix subspace is a vector space formed by linear combinations of vectors associated with a matrix

Trace

Sum of diagonal elements

Determinant

Scaling factor in a linear transformation (ad - bc in 2×2)

Invertible

There exists another matrix that can be multiplied with it to yield the identity matrix, indicating that the system of linear equations it represents has a unique solution

Rank

Represents the dimension of the vector space spanned by its rows or columns, indicating the maximum number of linearly independent row or column vectors in the matrix

Basis

A set of vectors that are linearly independent and span the entire vector space, meaning any vector in the space can be represented as a linear combination of these basis vectors

Span

Collection of all possible linear combinations of those vectors. It essentially describes the entire space that can be reached using the combinations of that set of vectors